A practical constitutive equation covering a wide range of strain rates

A practical constitutive equation covering a wide range of strain rates

OOZO-722517910901-09971$02.W0 A PRACTICAL CONSTITUTIVE EQUATION COVERING A WIDE RANGE OF STRAIN RATES SHINJI TANIMURA Department of Mechanical Engine...

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OOZO-722517910901-09971$02.W0

A PRACTICAL CONSTITUTIVE EQUATION COVERING A WIDE RANGE OF STRAIN RATES SHINJI TANIMURA Department of Mechanical Engineering, University of Osaka Prefecture, Sakai, Osaka 591. Japan Abstract-A practical constitutive equation of a single form combining stress, strain, strain rate and temperature effect is proposed with which the various material behaviors can be described covering a wide range of strain rates, espeeiaily the high strain rates. It is ~1~s~~ that the proposed form of the equation is useful when compared to some experimental results obtained under a dynamical uniaxial stress and an incremental torsional impact. 1. INTRODUCTION MANYTYPESof

constitutive equations, taking the time dependence into consideration, have been proposed, The constitu~ve equation for the elastic~vis~oplasti~~plasticbody has been proposed by Cristescu[l, 21. The constitutive equation emphasizes both instantaneous and non-instantaneous plastic properties which have been friquently under discussion in the dynamic plasticity field. In special cases, the equation reduces to the forms of the classic Prandtl-Reuss constitutive equation, the Perzyna type constitutive equation and the Hohenemser-Prager constitutive equation. It has been shown by Campbell and Dowling[3] that the incremental wave velocities almost coincided with the elastic wave velocities through the experiment in which the incremental torsional impact was applied to a tubular specimen. Even if the instantaneous plastic property exists, the experimental result is indicative that the component is indeed little. As a result, one can say that the elasticlviscoplastic constitutive equation represented by the Perzyna type may be practically applied. By studying the deformation mecha~sms of ~cros~c~~s, on the other hand, the Johnson~ilman type constitutive equation has been proposed reasonably for LiF crystals and b.c.c. metals, and the mechanism of the cutting of dislocations and the mechanism of cross slips have attracted attention for the deformation of f.c.c. metals. It has ben conjectured that, at a very high strain rate region above ld set-’ where the deformation stress increases rapidly, the phonon viscosity might play a dominant role. A practical form of the generalized constitutive equation has been proposed by Lindholm[4] which was developed from the ~errn~ly-activated dislocation model of deformation and applied to some experimental results for aluminum. The phenomenon of the rapid increase of stress at the high strain rate region above 103see-‘, however, could not be represented by his equation. Kuriyama and Kawata[S] studied the propagation of stress wave in a thin rod, applying the Johnsto~~man type constitutive equation, and they pointed out that the forms of the constitutive equations of other metals than a mild steel might also bear a close resemblance to that of the Johnston-Gilman type. In this paper, a practical constitutive equation of a single form combining stress, strain, strain rate and temperature effect is proposed, with which the various material behaviors can be represented, covering the wide range of strain rate, especially the high strain rate above lo3 set-‘. It is i~us~ted that the experimental results for ~~nurn given by Lindholm[4] can be represented reasonably well by the proposed form. Through the experiment in which the incremental torsional impact was applied to a thin-walled tube specimen of aluminum, the propagation velocities of each incremental strain level were measured. It is also shown that the proposed equation is useful when applied to these experimental results. 2. GENERALIZED

CONSTITUTIVE

EQUATION

Most of the plastic deformation in crystalline solids is, as well-known, caused by the movement of dislocations. The relations between the gross deformation speed and the applied stress which have been obtained experimentally for polycrystalline metals of f.c.c., b.c.c. and UES Vol.

997 17, No. !+-A

SHINJI TANIMURA

k 20

-

Aluminum IlOOTens. c+ &

Camp. 9 P

Temp. 294’K

-----

Llndholm

-

cuicutet~

672O K

Stmin rata kc+) Fig. 1. Stress vs strain rate at constanttemperatureand strain. After Lindholm (1948).

the others show the rate dependency as shown in Fig. 1. The relation between the dislocation velocity and the applied stress in single crystals has also been examined for many materials, and many experimental results as shown in appended Fig. 1A have been presented. In the comparatively low speed deformation range of these experimental results, the dislocation motions must be subjected to the thermal activation process. In the higher speed deformation range, the frictional force, due to the radiation and the scattering of phonons, may take the larger part of the deformation stress (see Appendix B). In the low speed deformation subjected to the thermal activation process, the strain rate for a simple shear may be expressed in the following form on the basis of the dislocation theory 29’ = Nv,b exp (- V/U)

(1)

where i.p is an inelastic component of shear strain rate (tensor definition), T is absolute temperature, N is movable dislocation density, v, is the elastic shear wave velocity ( = cd, b is the Burgers vector of the average dislocation, V is activation energy, and k is Boltzmann constant. Though eqn (1) is applicable for LiF crystal or b.c.c. metals from the tlrst, the same form may be applied formally to the deformation of f.c.c. metals which have a small PeierlsNabarro force. For the deformation mechanism of f.c.c. metals, we consider Nv,b in eqn (1) to be equivalent to the frequency parameter. It may be assumed that the following form of V, which can approximately represent the actual phenomena for the comparatively low speed deformation subjected to the thermal activation process, can be expressed for T,,,2 T L 7*’

(2) where T is shear stress, 7*(jp, T) is the back stress due to the work hardening, and 5” is the integrated inelastic shear strain defined by +p = JilVl dt’ where t’ is the time variable and t denotes current time. The symbol r,,, denotes the upper limit stress in the deformation region subjected to the thermal activation process (see Appendix B). The symbol T*’denotes the lower limit stress under which eqn (2) can be used, and so 7*’ > T*. In a practical sense, T*’= 7* may be held. In the higher speed deformation, on the other hand, it has been observed that for many materials the deformation stress becomes rapidly greater with the higher dislocation velocity. As one of the possibilities for explaining these high speed phenomena, the following mechanism can be deduced noticing the kinetic energy of moving dislocation. When we consider the case of the screw dislocation moving with a high speed, for simplicity, the total energy of the dislocation per unit length may be given by eqn (Al) (see Appendix A), and it may become infinitely large when the speed becomes highe; and tends toward the elastic transverse wave speed. When one denotes the stress ~6 which is necessary to bring about the acceleration i on

A practicalconstitutiveequationcoveringa wide rangeof strainrates

999

the moving dislocation of the velocity u, the relation between these can be expressed approximately as follows (3) where ?d is the component of shear stress which is to be the source of the kinetic energy, E,, is the strain energy per unit length of the stationary screw dislocation (see Appendix A). Now, it can be easily supposed that, in an actual crystalline solids, the movement of dislocation may con~ue dearly with a partial acceleration motion, by being caught and passing through various kinds of obstacles such as impurities, other dislocations or others. Therefore, when one considers the dislocation motion which is moving with an average acceleration, the relation between the mean velocity of the motion and the mean stress may be expressed by a form shown as eqn (3). As a rough approximation, if we exchange the index 2/3 for I in eqn (3), we can interpret that the denominator and the nume~tor on the right side of eqn (3) are co~es~n~ng formally to those of eqn (2), respectively. The deformations in actual polycrystalline metals might be caused by very complex mechanism. For combining the two different mechanisms discussed above which are divided into the low and high speed deformation ranges, we can attempt to make a constitutive equation, as a practical, usable form, which can describe the various phenomena for usual polycrys~line metals covering the very wide range of strain rate. Then we assume the following form applicable in the wide range of strain rate, which can be formally derived from eqns (1) and (2) by putting D’/kT = 0, for 7 2: r* - D(fp, T) 2+p = N(Y%JJ exp 7 _ 7*(p Tj ( , >

(4)

where D(fP, T) is a parameter which exhibits the strain rate sensibility. We regard that the inviscid plastic component is, in practice, negligible in comparison with the viscoplastic component in the inelastic strain component, and assume the isotropic hardening. Extension of eqn (4) to general states of stress can be made in a manner similar to that of Lindhotm143. The following equation can then be obtained as a practical form of the generalized constitutive lotion

(9

where 4 is the deviatoric strain rate tensor and the superscriptp denotes the plastic (inelastic) part, ep is the integrated plastic strain defined by zp = J{IZ$I” dt’ where I!$ = (l/2)++& Z,‘*’is the second invariant of the deviator& stress tensor SQe,,, = 43, a,,, = U&3,G is the shear modulus, K is the bulk modulus and a is the coefficient of thermalexpansion. This eqn (5) can be regardedas one of the practicalforms modifiedfrom the Perzyna type [6], under constant temperature, and one of the extended forms of the Malvem type in the treatmentbased on the non-instantaneous plastic property. It can be regardedthat the eqn (5) is the extended form of the generalized states of stress by applying a form, resembling the Johnstoffiilman type and the type used by Kuriyama and Kawata[5], to the wide range of deformation speed. The dependence of the yield function on strain rate and temperaturecan be examined by

J

(I$!) = i N(BP)v,b exp (

- D(Sp, T) 7*(‘, Tx~/I:2’/~*(p, T) - 1)>*

(6)

From eqn (6), if t/frS2) --*+ 00,t/Cl& + Numb/2and Nv,b/t denotes the upperlimit of plastic strain rate under the deformed state.

SHINJITANIMURA

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For more practical forms of N and tt, the following forms may be used

N(P) = N*+ CP

(7)

D(P, T) = l&(T) + H(T) (P)”

03)

where NOand C are material constants, independent of temperature, and n denotes the index of work hardening. It may be perceived that, for f.c.c, metals, the ~nsjt~~ity of the change of moveabfe dislocation density with plastic deformation is ~ornp~a~~e~ysmali, so that C = 0 may approximately be assumed for f.c.c, metals. From many experimental results, presented up-to-date, which examined the rate dependency on the deformation stress, it can be concisely concluded that the rate dependency for f.c.c. metals is not remarkablein the emah strain region but becomes remarkable with iarger strain; on the other hand, the dependency for b.c.c. meta& is conspicuous near the yield point and reduces to less with larger strain. From these, it may approx~ately be assumed that & = 0 for f,c.c. metals and H < 0 or H = 0 for b.c.c. metals. 3.COMPARISONWITH

SOME

EXPERIMENTALRESULTSANDDISCUSSION

(a) Uni-axial stress state fn this case, eqn (% reduces to

where o‘ and e are stress and strain in the axial direction and i’ = 0 is assumed. When denoting P = ~~3~2)~p where ZB = 14]@]dt’ and using eqns (7) and (8) for the forms of N and 9 eqn (9)

becomes & = P((I/3/2)i”, T) -

Z&(T)+ H(T){(I!3/2)P)} In ((~33iP/u,b)/{ni,+ (V3/2)Cgp})’

(10)

In Fii. I, the exigent restits for ~~~~ given by ~d~~rn~4] are shown. Broken lines in the figure are the app~~~t~ straight lines by the constitutive equation proposed by Lindholm and full lines are the calculated results by eqn (9). In the calculation, D = 21.5 ksi and 33.6 ksi and I* = 6.23 ksi and 0 were chosen at T = 294 and 672”K, respectively, and M&/2 = 106sec-’ was chosen. In this case, C = &(T) = 0 can be assumed, and H(T) a T is appro~~tely held. (In the case of the experimental rest&s for upturn given by I&user, et alJ71, C = I>dT) = 0 and n = l/2 can be assumed, but the value of H(T) is huge at a Iower temperature.) The static stress under a subsequent loading for a dynamically deformed state does not always coincide with the stress on the initial curve. For a material undergone a work hardening, it may approximately be able to substitute the static stress in the subsequent simple shear for the value of @. According to the survey of rate dependent strength properties of metals by ~~ho~rn and Bessey[8], it has been shown, for several ahtminum ailoys, that the value of D may remain almost can&ant with temperature or become somewhat larger at a higher temperature and for several low carbon steel the value may also remain almost constant with temperature. Therefore, if, for f-c-c. metals, C = 4 = 0 can be used, H(T) a T may be approximately held and if, for bc.c. metals, H = 0 can be used, B&P) may become a material constant i~e~ndent on temperature. (b) Torsion of a thin-wailed tube In this case, eqn (5)t reduces to

where y (tensor definition) is shear strain and eqns (7) and (8) are used.

1001

A practical constitutive equation covering a wide rahge of strain rates

We consider the case in which an incremental shearing impact is applied with a constant strain rate of 29 = 2j, (= const.) to the thin-walled tube which is being given a plastic strain 2yP statically. In this case, eqn (11) becomes

;!&+J$-

c

(No + C+P)ffmb exp

(

-PO(T) + H(T)(?P)“] 7 - ?*(Tp, T) >

(12)

and the propagation velocity c$. of each level of the incremental strain is given by c$= {(dT/dy)/2p}‘” where p is the density. By putting c2 = d/G/p, the last equation becomes c;.Ic2 = {(d7/dr)/2G}“2.

(13)

From eqn (13), for A? 4 y we can write as follows r=r,+G

I0

=Y (cM2(2dy)

(14)

where 7s denotes the static shear stress at pre-strain 2y and 2Ay is the incremental strain. When T* = 7,, the following relation can then be obtained by eqns (12)-(14) I_

(cf&32

=

wo+

CfPb=JJ

exp

-mm (

2%

+

~m(fPn

>

G o ’ (cYc3*(2dy) Y

(15) *

If we assume the following form[9] of the constitutive equation instead of eqn (11)

(16)

9 = +/2G + p[exp (~‘(7 - ‘r*)/r,} - 11, the following relation can be derived 1 - (c;I’cd2 = f (exp {$lUy

(cgc2)2(2du)] - I),

(17)

where a’ is a material constant (at a certain constant temperature), /3 = 742~~ 7 is the viscosity coefficient and 7. is the initial yield stress in static shear. The propagation velocities, c;i of each incremental strain level were measured through the experiments in which an incremental shearing impact was applied to a thin-walled tube specimen. The tubular specimens used are 12 mm in outer diameter, 10 mm in inner diameter, and 850 mm in length, made of solid drawn tube of commercial pure aluminuti, which annealed at 380°C during 2 hr. As shown in Fig. 2, an input bar and the tubular specimen were joined to each other by adhesive, Scotch-Weld 2214 (see Refs. [9, lo] for detail of the experimental apparatus and method). The values of c+ obtained by measuring a time difference between the arrival times of the same levels of incremental strain on two oscillograms at gage position Gl and G2 are shown in Fig. 3 by small circles. 1250

Input bar Fig. 2. Joint of input bar and tubular specimen.

lobular

spacimen

SHINJI TANMJRA

1002

-M+---------

E~~rimantal Eqn 115) Eqn (17)

_ (Pro-strain 0.22%)

ol

2

I

3

4xKI

26~

Fig. 3. Propagation velocities for each tevel of the incremental shear strain.

The full line in Fig. 3 shows the calculated results by eqn (15) choosing (Do + H(jP)“}/G = 3.86 x 10V5,C = 0 and ~~~~~/2~~= 5.59. The dotted line show the calculated results by eqn (U), choosing ar’Gfr0= 1000 and #Vi; = 32.8. From the figure, it may be seen that the characteristic of the experimental curve can be represented by eqn (15) fairly well. 4. CONCLUDING

REMARKS

To make the constitutive equation applicable to various kinds of actual polycrystalline metals, a practical form of the generalized ~ons~tutive equation combin~g stress, strain, strain rate and temperature effect has been proposed tentatively. It is illustrated that the proposed constitutive equation can represent well some phenomena for a wide strain rate range covering a high strain rate and is applicable not only to b.c.c. metals but also to f.c.c. metals. However, it may be easily supposed that the microscale deformation mechanism must be varied with crystal structures, deformation speeds and temperature and is very complicated, so that even if those phenomena can be we~desc~~d un~catively, it is nothing but a resemblance of those various phenomena with the practical form. It is expected that the physically based approach for the understanding of crystal plasticity and the phenomenological approach are combined more than at the present for the development to represent the constitutive behaviors of materials. The experimental results in the high strain rate range of about ld set-’ or above are yet comparatively few. For those experimental results, it may sometimes be supposed that there are some deputies about ex~~rnen~ accuracy and the ~ssib~ity of mixing the accelerating stress in the material strength, so that sometimes a considerable difference between the experimental results by investigators can be seen. So, more experimental results in the high strain rate range may be expected to be obtained further. Ac~owfe~g~@~~-~e

author wishes to thank Prof. Dr. H. Igaki for his support during this investigation.

REFERENCES [l] N. CRISTESCLJ,Dynamic Flasficity. North-Holland, Amsterdam (1%7). [2] N. CRI!$TBSCU,Mechanical Bduruior of Materid~ under Dynamic Loads, p, 329. Springer-Verlag, Berlin (1%8). [3] J. D. CAI@BELL and A. R. DOWLING. L Me& PITYJ.soiidt, 18.43 (1970). [4] U. S. LI~HU~, Meci&c~ &krrosrr & ~~~ wdrr J&+&c Loads, p. 77. Spy-Verl~, Berlin (196%). [S] S. KURIYAMA sod K. KAWATA, 3. Appl. phys. 44,3445 (1973). [6] P. PERZYNA, Q. A@. Moth. W, 321(1%3). [7] F. E. HAUSER, J. A. SIMMONS and J. E. DORN, Response of Metals to High Vefociry Deformorion, p. 93. Interscience, New York (l%l).

A practical constitutive equation covering a wide range of strain rates

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(81 U.S. LINDHOLM and R. L. BESSEY, Tech. Rept. AFML-TR-69119, Air Force Materials Laboratory, WrightPatterson Air Force Base, Ohio (1%9). [9] S. TANIMURA, H. IGAKI. H. MAJIMA and M. TADA, BrdIerinof Ihc ISME, 21, 1455(1978). [lo] S. TANIMURA, H. IGAKI and U. MORI, &print of the Lectm Meting of Japan Sm. hfech. Engrs. (in Japanese), No. 774-L D. 25 (1972). [II] T. NINOMIYA and S.‘ISHIOKA, 1. Phys. Sm. Japan, 23, 361 (1%~). (Received 30 November 1978) APPENDIX A: STRESS UNDER HIGH SPEED MOTION OF DISLOCATION The total energy E per unit length of a moving screw dislocation in an isotropic elastic body is given by E = E&l - uYc)“2

(Al)

where E0 is the strain energy per unit length of the stationary screw dislocation, and cr is the elastic transverse wave velocity. The kinetic energy is given by E - Er,, and we define the momentum p as follows in reference to the moving direction of the dislocation fd =

dp/dt = d(E

-

Eo)/dx

(A2)

where fd is the acting force per unit length of the dislocation, x is the axis chosen in the moving direction and I is time. In the case given by eqn (Al), by using the relation dx = u dr and the initial condition p = 0 at II = 0, the momentum p becomes p = (E/c30

(A3)

and fd becomes fd = (Edc)(l

2 3/zir - u*IQ)-

(A4)

where d = duldf. When a change of d is applied to the screw dislocation which is moving with velocity u, the acting force fd in the moving direction of the dislocation is given by eqn (A4). It may be regarded that (&/C:)(l - u2/&“* denotes the effective mass per unit length of the moving dislocation. It has been shown, on the other hand, by Nimomiyaand Ishioka[l 11that the effective mass perunit length m&for stationary vibration is given by the relation m& = &/2c: for a fairly wide band of frequency. Therefore, when a screw dislocation accompanied with vibration is moving with average velocity u, it may be able to approximately express the relation by adding a constant (for example, a value between 0.5 - 2) on the right side of eqn (A4).When we denote the shear stress component by 76 which corresponds to the stress required to cause the kinetic energy, the following relation can be obtained from eqn (A4)and by Using fd = 7db

(u/d2= 1- (&,ti/C:hd)213. When (u/c)* 4 I, eqn (As) can be written as follows U k C2 CXp(-{(EorilC:b)/7d}*"/2).

In v,

r Frictional forea r,

Actwl deformation

Fig. IA.

L46)

1004

SHINJI TANIMURA

It is shown in eqn (As) or (A6) that for a moving screw dislocation with average velocity I),the stress rd is proportional to the acceleration ir of the dislocation. It may also be understood, on the other hand, from these equations that to apply the same value of ir to the dislocation the larger value of r4 is needed when the value of u is larger. For the moving dislocation with comparatively hi speed, it can be interpreted that r,l is the stress component which is needed to supply the dissipating kinetic energy due to the phonon and similar effects. APPENDIX B: CONNECTION BETWEEN THE STATIC AND THE DYNAMIC DEFORMATIONS The largest part of the plastic (inelastic) deformation for the crystalline materials is attained, as well-known, by the dislocation motion. In the range of comparatively low speed deformation, the dislocation motions must be subjected to the thermal activation process. At a very high speed deformation, the frictional force (TV),due to the radiation and scattering of phonon, may take a large part of the dynamic stress. For connecting these processes for the low and the high speed deformations, the relation between the dislocation velocity and the shear stress may be simply explained schematically as shown in Appended Fig. IA. In the figure, u is the dislocation velocity and r* is the back stress due to the work hardening. The upper limit stress under which the deformation must mainly be subjected to the thermal activation process, may be denoted by r,,,, whose value may necessarily be varied by each deformation mechanism which must differ with each microscopic structure, and is changed by the effect of the impurities or the others. The symbol r, also implies the stress required to move the dislocation without any aid of the thermal vibrations for the various deformation mechanisms; for example, the mechanism in which the Peierls-Nabarro force may act dominantly, the mechanism in which the dislocation slips out of the interacted field formed with solute atoms or impurities, the mechanism of cutting dislocations or of cross slip, and others.